Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Relationship between trace and phase space

  1. Jun 18, 2013 #1
    So I noticed we can define entropy in two very different ways:
    1) quantum mechanically
    [itex]S = -k Tr(\hat{\rho}\ln{(\hat{\rho})}) [/itex]
    2) classically
    [itex]S = -k \int \rho \ln{(\rho)} d\Gamma[/itex]


    where Tr is the trace and [itex]d\Gamma = \frac{1}{h^{3N}N!}\prod_{i}^{N} d^{3N}q_{i}d^{3N}p_{i}[/itex] is the phase space volume element.

    My question is how summing over the diagonal of an operator is like integrating the same quantity over phase space.

    Edit: I realise that we are working out a mean value.
     
    Last edited: Jun 18, 2013
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted